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This book is based on results obtained over a decade of study and research. It questions the use of dynamic molecular models in the continuum scale providing alternative solutions to open problems in the literature. It provides a physical-mathematical understanding of the differential equations that govern fluid flow and energy transport, serving as a reference to the application of Smoothed Particle Hydrodynamics
in continuum fluid mechanics and transport phenomena. The physical-mathematical modelling of the problems in the continuum scale and the employment of the SPH method for solving the equations are presented. Examples of applications in continuum fluid mechanics with numerical results and discussions are also provided. This literature defends the concepts of continuum mechanics and the application of boundary treatment techniques that do not violate the laws of physics.
Smoothed Particle Hydrodynamics
A brief history of the Lagrangian modelling and Smoothed Particle Hydrodynamics method is presented. The advantages in relation to mesh methods, the growth of its use for the solution of the physical laws governing the flow of fluids and the transport of energy, with a view to the advancement of computational processing techniques, are presented. The importance and frequency of its use in solving problems in the continuum mechanics are also emphasized. Finally, a brief presentation of the contents of each subsequent chapter is presented.
2 Physical-Mathematical Modelling
Chapter 2 presents the continuum hypothesis that enables that the physical laws of conservation (mass, momentum and energy) be written by mathematical partial differential equations. Besides these equations, the equation of state for the prediction of the dynamic pressure of a fluid flowing, the concept of the modified pressure and the modelling of the internal energy are also presented.
2.1 The Continuum Hyphotesis
2.2 Physical Laws of Conservation
2.3 Pressure Modelling
2.3.1 Equation of State for Dynamic Pressure
2.3.2 Modified Pressure
2.4 Specific Internal Energy Modelling
3 Smoothed Particle Hydrodynamics Method
In this chapter, the fundamentals of the SPH method and its application in the discretisation of the continuum domain in particles are presented. The SPH approximations to the equations of conservation are deduced and explained. Kernels used in interpolations, temporal integration methods, the particle inconsistency problem and numerical corrections applied in simulations are also presented. A special attention is given to the presentation of commonly boundary conditions techniques applied in SPH. The use of virtual or dynamic particles and artificial repulsive forces are discussed and the reflective boundary conditions technique, which respects the laws of physics and continuum mechanics, is defended.
3.2 Discretisation of the Continuum Domain
3.2.1 Approximation of the Divergent of a Vectorial Function
3.2.2 Approximation of the Gradient of a Scalar Function
3.2.3 Approximation of the Laplacian
3.2.4 SPH Approximations for the Conservation Equations
3.2.5 Errors in SPH Approximations
3.2.6 Smoothing Functions
3.2.7 Neighbouring Particle Search
3.2.8 Treatment of the Free Surface
3.2.9 Treatment of the Interfaces